To analyze the effects of excluding an outlier on the median, mean, mode, and range of the dataset \(5, 10, 12, 14, 18, 24, 60\), we first need to identify the outlier in the set.
Step 1: Identify the Outlier
In this series, \(60\) is significantly higher than the other numbers, which suggests it is the outlier.
Step 2: Calculate Statistics with the Outlier
With the Outlier (60):
- Data Set: \(5, 10, 12, 14, 18, 24, 60\)
-
Median:
- To find the median, arrange the data in order (which is already done).
- The number of data points (n) is 7 (odd).
- Median = \( \text{middle value} = 14\) (4th value).
-
Mean:
- Mean = \( \frac{(5 + 10 + 12 + 14 + 18 + 24 + 60)}{7} = \frac{133}{7} \approx 19\).
-
Mode:
- Mode = There is no repeating value, so there is no mode.
-
Range:
- Range = Maximum - Minimum = \(60 - 5 = 55\).
Step 3: Calculate Statistics without the Outlier
Without the Outlier (60):
- Data Set: \(5, 10, 12, 14, 18, 24\)
-
Median:
- Now we have 6 data points (even).
- Median = \( \frac{(12 + 14)}{2} = 13\).
-
Mean:
- Mean = \( \frac{(5 + 10 + 12 + 14 + 18 + 24)}{6} = \frac{73}{6} \approx 12.33\).
-
Mode:
- Still, there is no repeating value, so the mode remains non-existent.
-
Range:
- Range = Maximum - Minimum = \(24 - 5 = 19\).
Summary of Changes
-
Median:
- With outlier = 14; without outlier = 13 (decreased by 1).
-
Mean:
- With outlier = 19; without outlier ≈ 12.33 (decreased by approximately 6.67).
-
Mode:
- With outlier = None; without outlier = None (remains unchanged).
-
Range:
- With outlier = 55; without outlier = 19 (decreased by 36).
Excluding the outlier \(60\) lowers the median, mean, and range, while the mode remains unchanged.
1 answer